
If \({\alpha }_s(e) < {\alpha }_m(e)\) , we have \(\xi _{{\alpha }_{m}(e),e}(2\pi ) \approx N_1(-1,1) \diamond R({\theta })\)  for some \({\theta }\in (\pi ,2\pi )\) . Thus it is spectrally stable and linearly unstable.

 If \({\alpha }_k(e)<{\alpha }_s(e) ={\alpha }_m(e)\) , we have \(\xi _{{\alpha }_{s}(e),e}(2\pi ) \approx -I_2 \diamond R({\theta })\)  for some \({\theta }\in (\pi ,2\pi )\) . Thus it is linearly stable, but not strongly linearly stable.

If \({\alpha }_k(e)<{\alpha }_s(e) < {\alpha }_m(e)\) , we have \(\xi _{{\alpha }_{s}(e),e}(2\pi ) \approx N_1(-1,-1) \diamond R({\theta })\)  for some \({\theta }\in (\pi ,2\pi )\) . Thus it is spectrally stable and linearly unstable.

 If \({\alpha }_k(e)<{\alpha }_s(e) \le {\alpha }_m(e)\) , we have \(\xi _{{\alpha }_{k}(e),e}(2\pi ) \approx N_2(e^{\sqrt{-1}{\theta }},b)\)  for some \({\theta }\in (0,\pi )\)  and \(b= \begin{pmatrix}b_1 & b_2 \\ b_3 & b_4 \end{pmatrix}\) 
satisfying \((b_2-b_3)\sin {\theta }> 0\) ,
that is, \(N_2(e^{\sqrt{-1}{\theta }},b)\)  is
trivial in the sense of Definition 1.8.11 in p.41 of [1]}.
Consequently the matrix \(\xi _{{\alpha }_{k}(e),e}(2\pi )\)  is spectrally stable and linearly unstable.

If \({\alpha }_k(e) = {\alpha }_s(e) < {\alpha }_m(e)\) ,
we have either \(\xi _{{\alpha }_{k}(e),e}(2\pi ) \approx N_1(-1,1) \diamond D({\lambda })\)  for some \(-1\ne {\lambda }<0\) 
and is linearly unstable; or \(\xi _{{\alpha }_{k}(e),e}(2\pi ) \approx M_2(-1,c)\)  with \(c_1\) , \(c_2\in {\bf R}\)  and \(c_2 \ne 0\) . Thus it is spectrally stable and linearly unstable.

If \({\alpha }_k(e) = {\alpha }_s(e) = {\alpha }_m(e)\) , either \(\xi _{{\alpha }_{k}(e),e}(2\pi ) \approx M_2(-1,c)\) 
with \(c_1\) , \(c_2\in {\bf R}\)  with \(c_2\ne 0\) ,
or
\(\xi _{{\alpha }_{k}(e),e} \approx N_1(-1, 1)\diamond N_1(-1, 1)\) .
Thus \(\xi _{{\alpha }_{k}(e),e}(2\pi )\)  is spectrally stable and linearly unstable.


